Italy's Jakob Weger thrilled his home crowd on the Ivrea course by winning gold in 77.22s, just 0.03s in front of Bradley Forbes-Cryans, of Great Britain, with Sweden's Eric Holmer third in 80.85s.
Butcher was just 0.04s behind after his 80.89s run, crediting his huge support team - which includes Olympic champion triathlete Hamish Carter - and a ton of hard work for his breakthrough performance.
"I'm just stoked that a lot of work came together for a moment that counted. I've been putting in a lot of work in New Zealand and not just in training - with my strength and conditioning coach, with my sports psychologist, with my nutritionist and with my coaches.
"I'd talked to Hamish about trusting the buildup and knowing I'd done all the work, it was all behind me and I'd done everything I could to prepare was pretty sweet."
Mutton, who was eighth in the junior final last year, also qualified 10th after a nerve-wracking semifinal.
"I was pretty stoked just to be in the final because my semifinal was not good - I was pretty terrified actually. I was shaking a fair bit on the start line and I didn't execute the run all that well but managed to slide through in 10th spot."
He managed to turn it all around in the final, one of just four paddlers to dip under the 80s barrier. Crucially, however, he had a slight touch on the seemingly innocuous Gate 5, picking up a 2s penalty to drop his overall time to 81.84s.
"After the semifinal, I got it under control, felt like a new person, and went sub-80 in the final, which I was really happy with. It turned into an 81 with my touch and it was a stupid touch, which sucks, but it was a really nice run and it was a good time.
"Overall, I'm pretty gutted but hopefully I'll be able to take some positives out of it soon."
Two other New Zealanders were in action on the final day of the championships, with Hannah Thomas finishing 19th in her junior C1 semifinal and 15-year-old George Snook knocked out in the quarter-finals of the junior extreme slalom.
Swiss paddler Jan Rohrer, who spent last year in Rotorua on an exchange training with Mutton and Snook, won the extreme slalom final.